On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, and Ludwig Reich. "On Probability Distribution Solutions of a Functional Equation." Bulletin of the Polish Academy of Sciences. Mathematics 53.4 (2005): 389-399. <http://eudml.org/doc/281092>.

@article{JanuszMorawiec2005,
abstract = {Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, $φ|_\{(-∞,0]\} = 0$, $φ|_\{[1,∞)\} = 1$, = φ: ℝ → ℝ|φ is continuous, $φ|_\{(-∞,0]\} = 0$, $φ|_\{[1,∞)\} = 1$. We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection between solutions in both classes.},
author = {Janusz Morawiec, Ludwig Reich},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {functional equations; increasing solutions; continuous distribution functions},
language = {eng},
number = {4},
pages = {389-399},
title = {On Probability Distribution Solutions of a Functional Equation},
url = {http://eudml.org/doc/281092},
volume = {53},
year = {2005},
}

TY - JOUR
AU - Janusz Morawiec
AU - Ludwig Reich
TI - On Probability Distribution Solutions of a Functional Equation
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2005
VL - 53
IS - 4
SP - 389
EP - 399
AB - Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, $φ|_{(-∞,0]} = 0$, $φ|_{[1,∞)} = 1$, = φ: ℝ → ℝ|φ is continuous, $φ|_{(-∞,0]} = 0$, $φ|_{[1,∞)} = 1$. We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection between solutions in both classes.
LA - eng
KW - functional equations; increasing solutions; continuous distribution functions
UR - http://eudml.org/doc/281092
ER -